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Price Variation Limits              and Financial Market Bubbles:               Artificial Market Simulations             ...
Introduction    restrict trades                                           market more                       Price Variatio...
restrict trades                                      market more                        Price Variation Limit             ...
Artificial Market Model (Agent Based Model)    Chiarella et. al. [2009]     ● Continuous Double Auction         ⇒ to imple...
restrict trades                                       market more                      Price Variation Limit              ...
Experiment 1: Learning and Overshoot          Case 1     Fundamental Value         = constant                             ...
Case 1: Fundamental Value = constant                                                         case 1                  10100...
Case 2: Fundamental Value → rapid increment (bubble trigger)                                                       case 2 ...
Traditional Stylized Facts                                       case 1                         case 2                    ...
Hazard Rate (similar to “run test”)      New Stylized fact to verify model replicating overshoot      H(i) conditional pro...
New Stylized Facts: Hazard Rate H(i)                                     case 1                      case 2               ...
Result Summary Experiment 1: Learning and Overshoot                            Not-Consistent with Consistent with        ...
restrict trades                                         market more                      Price Variation Limit            ...
Experiment 2: Price Variation Limit   Price Variation Limit                                                               ...
Case 2 & with Learning, Price Variation Limit                                                   case 2, learning          ...
Overshooting Amplitude and Converging Speed                                                             case 2, learning  ...
Max Price from New Fundamental Value    max price from new                                            tpl     fundamental ...
Reach time to New Fundamental Value     reach time to new     fundamental value                                           ...
Result Summary Experiment 2: about Price Variation Limit      Price Variation Limit            prevent Overshooting and Co...
Summary    restrict trades                                           market more                       Price Variation Lim...
That’s all for my presentation.                  Thank you very much                  for your cooperation !              ...
Appendix           22                22
Artificial Market Model (Agent Based Model)       On basis of Chiarella et. al. [2009]         + Learning Process of agent...
Agent Model    j: agent number (1000 agents)                     Historical Return                                        ...
Learning Process       Expected Return                              1                     P                              ...
Order Price and Buy or Sell                                    price        Pet, j + Pd                            P   t  ...
Hazard Rate                                         case 2 lerning                                 non Price Limit Price L...
Strategies Weight                                                                    case 2, learning                     ...
Strategies Weight                                                         case 2, learning, Price Limit                   ...
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Price Variation Limits and Financial Market Bubbles: Artificial Market Simulations with Agents' Learning Process

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Price Variation Limits and Financial Market Bubbles: Artificial Market Simulations with Agents' Learning Process: presented by Takanobu MIZUTA et. al., EEE Symposium Series on Computational Intelligence, Computational Intelligence for Financial Engineering and Economics (CIFEr), 2013.

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Price Variation Limits and Financial Market Bubbles: Artificial Market Simulations with Agents' Learning Process

  1. 1. Price Variation Limits and Financial Market Bubbles: Artificial Market Simulations with Agents Learning Process SPARX Asset Management Co. Ltd. Takanobu MIZUTA* The University of Tokyo Kiyoshi IZUMI The University of Tokyo CREST PRESTO, JST Shinobu YOSHIMURA University of Tokyo * mizutata@gmail.com * http://www.geocities.jp/mizuta_ta/ 1 http://www.slideshare.net/mizutata/cifer2013Thank you very much. I’m Takanobu MIZUTA from SPARX Asset Management.I’m also belonging to The University of Tokyo.Today, I’m going to give a presentation under the title of This. 1
  2. 2. Introduction restrict trades market more Price Variation Limit ? efficient or not? out of certain price ranges to Prevent Overshoot Experiment 1 Experiment 2 Replicate Overshoot? Parameters’ Artificial Market Model Condition Verified by ⊿Ppl, tpl Hazard Rate (Agent Based Simulation) NEED NEED Learning Process ∆Ppl / t pl < v Should Avoid ∆Ppl / t pl << v 2 http://www.slideshare.net/mizutata/cifer2013■ Financial exchanges sometimes employ a “price variation limit”, which restricttrades out of certain price ranges to avoid sudden large price fluctuations,“overshoots” overshoots occur in bubble and crash in real financial market■ There is a debate over whether the price variation limit makes financial marketmore efficient or not.■ To investigate price variation limits, we built an artificial market modelimplementing a learning process.■ Experiment 1, we will show that the model should be implemented the learningprocess to replicate overshoot.■ Experiment 2, We will also show that a parameters condition of the pricevariation limit to prevent overshoot. 2
  3. 3. restrict trades market more Price Variation Limit ? efficient or not? out of certain price ranges to Prevent Overshoot Experiment 1 Experiment 2 Replicate Overshoot? Parameters’ Artificial Market Model Condition Verified by ⊿Ppl, tpl Hazard Rate (Agent Based Simulation) NEED NEED Learning Process ∆Ppl / t pl < v Should Avoid ∆Ppl / t pl << v 3 http://www.slideshare.net/mizutata/cifer2013First, I will describe our artificial market model. 3
  4. 4. Artificial Market Model (Agent Based Model) Chiarella et. al. [2009] ● Continuous Double Auction ⇒ to implement realistic price variation limit ● Agent model is Simple ⇒ to avoid arbitrary result “Keep it short and simple” heterogeneous 1000 agents Expected Return wi , j 1  P  ret, j =  w1, j log ft + w2, j rht, j + w3, j ε tj    ∑i wi , j  P Strategy Weight  Fundamental Technical noise ↑ Different + Learning Process Our Original for each agent ↑ need to replicate an overshoot Good Performance Strategy wi , j is up 4 Bad Performance Strategy wi , j is downWe built an artificial market model on basis of Chiarella et. al. [2009].●Pricing mechanism is Continuous Double AuctionIt is not simple, but, we need to implement realistic price variation limit●Agent Model is Simple. This is to avoid arbitrary result, “Keep it short andsimple” principle.We think Artificial Market Models should explain Stylized Facts as Simply aspossible.There are heterogeneous 1000 agents. All agents calculate Expected Returnusing this equation,And, the strategy weights are different for each agent・First term is a Fundamental Strategy: When the market price is smaller than thefundamental price, an agent expects a positive return , and vice verse.・Second term is a technical strategy: When historical return is positive, an agentexpects a positive return, and vice verse.・Third term is noise.Chiarella’s model did not include Learning Process, however,We built Learning Process of agents, this is our Original.We showed that learning process need to replicate an overshoot.Agents are comparing Historical Return and Each Strategy’s Return.・When the strategy’s return and Historical Return are Same Sign, GoodPerformance Strategy, The strategy’s Weight is Up.・When the strategy’s return and Historical Return are Opposite Sign, BadPerformance Strategy, The strategy’s Weight is Down. 4
  5. 5. restrict trades market more Price Variation Limit ? efficient or not? out of certain price ranges to Prevent Overshoot Experiment 1 Experiment 2 Replicate Overshoot? Parameters’ Artificial Market Model Condition Verified by ⊿Ppl, tpl Hazard Rate (Agent Based Simulation) NEED NEED Learning Process ∆Ppl / t pl < v Should Avoid ∆Ppl / t pl << v http://www.slideshare.net/mizutata/cifer2013 5Next, I show simulation results of Experiment 1 about leaning process andreplicating overshoot 5
  6. 6. Experiment 1: Learning and Overshoot Case 1 Fundamental Value = constant With Learning Process Case 2 Fundamental Value × For Without → rapid increment Learning Process Each (bubble trigger) Cases 6We examined two Cases, Case 1, Fundamental Value is constant,Case 2, Fundamental value is rapid incremented like this. This is bubble inducingtrigger.For Each cases, we examined With learning process And WithOut learningprocess.Therefore, we examined four cases in all. 6
  7. 7. Case 1: Fundamental Value = constant case 1 10100 10050 10000 price 9950 9900 9850 non-learning learning 9800 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 time Small fluctuating around Fundamental Value 7This Figure shows time evolution of market prices in case 1, Fundamental Valueis constant.In both cases, With learning process and without learning process. the results arevery similar,The prices were small fluctuating around Fundamental Value, here, TenThousand 7
  8. 8. Case 2: Fundamental Value → rapid increment (bubble trigger) case 2 20000 18000 16000 price 14000 non-learning 12000 learning 10000 fundamental value 8000 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 time Only with learning process, Overshoot occurred 8This Figure shows time evolution of prices in case 2.Fundamental value was changed at this time, increased to New FundamentalValue, Fifteen Thousand.This is the bubble inducing trigger.Without Learning Process, Black line, Overshooting was not occurred.On the other hand, with Learning Process, Red line,the price went far beyond the new fundamental value.Only with learning process, Overshoot occurred. 8
  9. 9. Traditional Stylized Facts case 1 case 2 non-learning learning non-learning learning kurtosis 3.018 5.394 2.079 3.180 lag 1 0.134 0.125 0.219 0.325 2 0.101 0.105 0.164 0.293 3 0.076 0.087 0.133 0.274 autocorrelation 4 0.060 0.074 0.118 0.261 coefficient for 5 0.052 0.061 0.108 0.253 square return 6 0.040 0.054 0.100 0.247 7 0.036 0.048 0.092 0.241 8 0.030 0.045 0.087 0.237 9 0.026 0.039 0.082 0.238 All cases replicated: Fat Tail and Volatility Clustering 9This Table lists Traditional stylized facts in each case.In all cases, both kurtosis and autocorrelation for square returns for all i arepositive.This means that all cases replicate Traditional stylized facts: fat-tail and volatility-clustering. 9
  10. 10. Hazard Rate (similar to “run test”) New Stylized fact to verify model replicating overshoot H(i) conditional probability that sequence of positive return ends at i, given that it lasts until i. For Example i=3、H(3) Time ・・・・ 1 2 3 4 H(3): Probability of Return 4th Return Negative Empirical Studies: Any cases: H(i) < 50%, Overshoot period: H(i) decline with I rapidly McQueen and Thorley [1994], Chan et. al. [1998] ⇒ Overshoot returns tend to continue to be positive this tendency stronger continuing positive returns longer 10We propose Hazard Rate as New Stylized fact to verify model replicatingovershootHazard Rate Hi is conditional probability that sequence of positive return ends at i,given that it lasts until i.For Example i=3, H3 means like this.1st, positive return, 2nd, positive, 3rd positive,In this condition, H3 is probability of 4th return become negative.Empirical Studies showed that, Any cases, Hi for most of i are smaller than 50%And when including overshoot period, Hi decline rapidly with i,This show that the overshoot returns tend to continue to be positiveAnd this tendency stronger continuing positive returns longer 10
  11. 11. New Stylized Facts: Hazard Rate H(i) case 1 case 2 non-learning learning non-learning learning i 1 56% 55% 56% 55% 2 55% 52% 55% 50% 3 55% 50% 53% 45% 4 54% 49% 52% 40% hazard rate 5 54% 45% 48% 36% 6 53% 44% 45% 29% 7 52% 41% 40% 26% 8 52% 40% 35% 22% 9 53% 40% 30% 19% Only with Learning process → Verified by Hazard Rate And Only Case 2 with Learning ⇒ Replicating Overshoot 11This Table lists New Stylized Facts: Hazard Rate in each case.In case 2 with learning, hazard rate declined rapidly.This case can replicate a significant Overshoot like actual markets.On the other hand, the case without learning, hazard rate dose not declinedrapidly.The case can not replicate Overshoot.Case 1, without learning, Hazard rates are upper 50% for all i.This is Not consistent with empirical study.On the other hand, Case 1, with learning, Hazard rates for most of i are smallerthan 50%, even when price fluctuations are stable.This consistent with empirical study.Therefore, only cases with Learning Process were verified by Hazard Rate,and only Case 2 can replicate overshoot. 11
  12. 12. Result Summary Experiment 1: Learning and Overshoot Not-Consistent with Consistent with Empirical study Empirical study ↑ ↑ Without With Learning Process Learning Process Case1 Stable Stable Fundamental Value Not-Verified by Verified by = constant Hazard Rate Hazard Rate Case2 No-Overshoot Overshoot Fundamental Value Not-Verified by (Bubble & Crush) → rapid increment Hazard Rate Verified by Hazard Rate 12Result Summary Experiment 1 relationship between Learning process andreplicating OvershootThe cases With learning process, both case 1 and case 2, were Consistent withEmpirical study verified by Hazard Rate.And case 2 can replicate overshoot, bubble and crushThe cases Without Learning Process were Not consistent with Empirical studyNot verified by Hazard Rate. 12
  13. 13. restrict trades market more Price Variation Limit ? efficient or not? out of certain price ranges to Prevent Overshoot Experiment 1 Experiment 2 Replicate Overshoot? Parameters’ Artificial Market Model Condition Verified by ⊿Ppl, tpl Hazard Rate (Agent Based Simulation) NEED NEED Learning Process ∆Ppl / t pl < v Should Avoid ∆Ppl / t pl << v 13 http://www.slideshare.net/mizutata/cifer2013Next, I show Simulation Results about Price Variation Limit. 13
  14. 14. Experiment 2: Price Variation Limit Price Variation Limit t −t p l Can Not order Outside P ± ∆Ppl Two Constant t pl Limit time Span Parameters: ∆Ppl Limit price Range changed to Buy order above here t − t pl P + ∆Ppl ∆Ppl t − t pl t pl P t − t pl P ∆Ppl Market Price Pt changed to t − t pl Sell order under here P − ∆Ppl 14We modeled the price variation limit like this.There are two constant parameters.Tpl is a limit time span, and ⊿Ppl is limit price range.We referred market price Before tpl, Pt-tpl.and any agents can Not order OutSide from Pt-tpl - ⊿Ppl to Pt-tpl + ⊿PplConcretely,Any buy order prices above here, they are changed to this price.and any sell order prices under here they are changed to this price. 14
  15. 15. Case 2 & with Learning, Price Variation Limit case 2, learning 20000 18000 16000 price 14000 12000 non Price Limit Price Limit 10000 fundamental value 8000 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 time Overshooting was vanished in price variation limit But, price took longer to reach new fundamental price 15This Figure shows time evolution of prices in case 2 with learning,comparing the case implemented price variation limit and not implemented.Overshoot was vanished In the case implemented price variation limitHowever, price took longer to reach new fundamental price.In the case not implemented price variation limit, price took faster to reach newfundamental price.Implemented case, slower to reach new fundamental value, converging is slower. 15
  16. 16. Overshooting Amplitude and Converging Speed case 2, learning 20000 reach time to new 18000 max price from fundamental value new fundamental value 16000 price 14000 12000 non Price Limit Price Limit 10000 fundamental value 8000 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 time Preventing Overshooting and Immediate Converging → no market achieves both at once Optimization of t pl and ∆Ppl 16Next, I investigated relationship between Overshooting Amplitude andConverging Speed.I measured, here, reach time to new fundamental value, and here, max pricefrom new fundamental value.We want Preventing Overshooting and Immediate Converging.However, No market achieves both at once.Therefore, it is important to Optimize of these Two parameters. 16
  17. 17. Max Price from New Fundamental Value max price from new tpl fundamental value 1000 2000 3000 4000 7000 10000 15000 20000 25000 30000 40000 50000 100 1,374 475 285 205 115 76 --- --- --- --- --- --- 200 3,236 1,421 728 485 224 146 97 72 --- --- --- --- 300 3,450 3,020 1,378 900 387 233 134 106 90 86 --- --- 400 3,482 3,501 2,381 1,307 565 339 164 112 95 93 85 --- 700 3,347 3,470 3,366 3,311 1,395 734 363 211 120 94 90 83 ⊿Ppl 1000 3,494 1500 3,512 3,357 3,356 3,229 3,424 3,578 3,404 2,633 1,388 681 3,407 3,019 1,384 421 607 267 286 122 208 93 184 91 183 2000 3,454 3,595 3,334 3,317 3,682 3,493 2,408 1,094 580 452 411 418 2500 3,475 3,357 3,497 3,368 3,262 3,466 3,524 2,007 1,442 508 103 100 3000 3,436 3,443 3,598 3,431 3,268 3,330 3,365 2,921 1,602 1,190 980 931 4000 3,359 3,398 3,374 3,607 3,673 3,378 3,338 3,351 3,433 1,701 477 401 5000 3,556 3,467 3,317 3,509 3,440 3,162 3,486 3,311 3,320 3,231 1,345 104 ∆Ppl v: Converging Speed Blue area: < v ≅ 0.128 without price variation limit t pl Preventing Overshooting ∆Ppl / t pl smaller (to upper right) ⇒ Overshoot smaller NEED to prevent overshoots ∆Ppl / t pl < v 17This table lists Max Price from New Fundamental Value in various Tpl and ⊿PplBlue area, ⊿Ppl over Tpl is smaller than VV is a converging speed without price variation limit, approximately 0.128.As you see, in this area, max price small, this means preventing overshoot.This is smaller, to upper right area, overshoot are smaller.Therefore, we found that this is smaller than V is needed to prevent overshoot. 17
  18. 18. Reach time to New Fundamental Value reach time to new fundamental value tpl (x 1000) 1000 2000 3000 4000 7000 10000 15000 20000 25000 30000 40000 50000 100 55 104 157 216 382 555 --- --- --- --- --- --- 200 40 55 79 103 181 261 385 509 --- --- --- --- 300 39 41 55 70 120 171 257 342 425 506 --- --- 400 39 39 44 55 91 128 195 259 324 385 505 --- 700 40 39 40 39 55 75 113 148 185 229 302 371 ⊿Ppl 1000 1500 39 39 39 39 39 39 39 39 41 39 54 40 78 54 104 72 127 93 150 113 193 146 234 175 2000 39 39 40 40 39 39 43 56 71 85 106 126 2500 39 39 39 39 39 39 39 46 53 60 74 86 3000 39 39 39 39 40 39 39 40 48 56 70 79 4000 39 39 39 39 39 39 40 39 39 44 63 76 5000 39 39 39 39 39 40 39 39 40 40 40 47 ∆Ppl Blue area: < v ≅ 0.128 v: Converging Speed t pl without price variation limit ∆Ppl / t pl smaller (to upper right) ⇒ converging speed slower Should Avoid Not to be converging slower ∆Ppl / t pl << v 18This table lists Reach time to New Fundamental Value in various Tpl and ⊿PplBlue area, this is smaller than V.This is smaller, to upper right area, converging speed are slower.Therefore, we found that it this is too smaller than V is should be Avoid Not to beconverging slower. 18
  19. 19. Result Summary Experiment 2: about Price Variation Limit Price Variation Limit prevent Overshooting and Converging Slower Optimization of t pl and ∆Ppl ∆Ppl / t pl smaller ⇒ Overshooting smaller NEED to prevent overshoots ∆Ppl / t pl < v ∆Ppl / t pl << v Should Avoid Not to be converging slower ∆Ppl / t pl smaller ⇒ converging speed slower v: Converging Speed without price variation limit 19Result Summary Experiment 2 about Price Variation Limit.Price Variation Limit prevents Overshoot and cause Converging speed Slower tonew fundamental valueTherefore, it is important to Optimize of these two parameters.When this is smaller, overshooting smaller.We found that it needs smaller than, V to prevent overshootsOn the other hand, this is smaller, converging speed is slower.We found that it should be avoid too smaller than V Not to be converging slower 19
  20. 20. Summary restrict trades market more Price Variation Limit ? efficient or not? out of certain price ranges to Prevent Overshoot Experiment 1 Experiment 2 Replicate Overshot? Parameters’ Artificial Market Model Condition Verified by ⊿Ppl, tpl Hazard Rate (Agent Based Simulation) NEED NEED Learning Process ∆Ppl / t pl < v Should Avoid ∆Ppl / t pl << v 20 http://www.slideshare.net/mizutata/cifer2013I summarize this presentation■ To investigate price variation limits, we built an artificial market modelimplementing a learning process.■ Experiment 1, we showed that the model should be implemented the learningprocess to replicate overshoot.■ Experiment 2, We also showed that a parameters condition of the pricevariation limit to prevent overshoot. 20
  21. 21. That’s all for my presentation. Thank you very much for your cooperation ! 21 http://www.slideshare.net/mizutata/cifer2013Could you say that again? (もう一度、おっしゃっていただけますか?)I don’t quite understand your question. (ご質問の趣旨が良く分からないのですが)Could you please rephrase your question? (ご質問を分かりやすく言い換えていただけますか)So, you are asking me about.... (つまり、お尋ねの内容は...ですね)I totally agree with you. (私も全くあなたと同意見です)That’s a very challenging question for me to answer. (それは私にとって非常に答えがいのある質問です)That’s a question I’m not sure I can answer right now. (そのご質問にすぐお答えできるかどうか分かりません)It would require further research. (さらなる研究結果を待ちたい)You are right on that point. (その点に関してはあなたが正しい)Our method will not solve the problem. (我々の方法ではその問題は解決できない) 21
  22. 22. Appendix 22 22
  23. 23. Artificial Market Model (Agent Based Model) On basis of Chiarella et. al. [2009] + Learning Process of agents comparing between the case with Learning Process and without Feature of our model ○ agent model is Simple → to avoid arbitrary result “Keep it short and simple” Models should explain stylized facts as simply as possible ● pricing mechanism is Continuous Double Auction → not simple to implement realistic price variation limit ☆ Learning process → agents switch strategy, fundamental or technical An overshoot occurred in the case with the learning process but did not occur in the case without the process 23We built an artificial market model on basis of Chiarella et. al. [2009].Chiarella’s model did not include Learning Process, however,We built Learning Process of agents.And we are comparing between the case with Learning Process and without it.Our Agent Model is Simple. This is to avoid arbitrary result, “Keep it short andsimple” principle.We think Artificial Market Models should explain Stylized Facts as Simply aspossible,Our pricing mechanism is Continuous Double AuctionIt is not simple, but, we need to implement realistic price variation limitLearning processHere, Learning process means agents are switching strategy, fundamentalstrategy or technical strategy.We will show that, an Overshoot occurred in the case With the learning process,however, overshoot did not occur in the case WithOut the process 23
  24. 24. Agent Model j: agent number (1000 agents) Historical Return t −τ j ordering in number order rht, j = log( P t / P ) t: tick time Technical Expected Return 1  P  ret, j =  w1, j log ft + w2, j rht, j + w3, j ε tj    ∑i wi , j  P  Parameters for agents Fundamental noise wi , j and τj Pf Fundamental Price Random of ε tj 10000 = constant Random of Normal Uniform Distribution P t Market Price at t Distribution Average=0 i=1,3: 0~1 σ=3% wi , j Expected Price i=2: 0~10 τj 0~10000 Pet, j = P t exp(ret, j ) 24Next, I will describe agent model.All agents calculate Expected Return using this equation.First term is a Fundamental Strategy:When the market price is smaller than the fundamental price, an agent expects apositive return , and vice verse.Second term is a technical strategy: When historical return is positive, an agent expects a positive return, and viceverse.Third term is noise,After the expected return has been determined, an expected price is determinedlike this.And, agents order base on this Expected Price. 24
  25. 25. Learning Process Expected Return 1  P  ret, j =  w1, j log ft + w2, j rht, j + w3, j ε tj    ∑i wi , j  P  Historical Return Compare each Strategy t −t l rl = log P / P t t Same Sign Opposite Sign good performer Strategy bad performer Strategy wi , j ← wi , j + krlt [0, ( wi , max − wi , j )] wi , j ← wi , j − krlt [0, wi , j ] Weight Up Weight Down With 1% probability: [a,b]: Random Uniform Distribution wi , j ← [0, wi ,max ] Reset from a to b 25We also developed a model implementing a learning process of agents.Agents are comparing Historical Return and each Strategy’ term, Fundamentalstrategy term, and Technical strategy term.When the strategy’s expected return and Historical Return are Same Sign,This means good performer Strategy.The strategy’s Weight is Up.When the strategy’s expected return and Historical Return are Opposite Sign,This means bad performer Strategy.The strategy’s Weight is Down.We also added random learning.In this way, agents learn better parameters and switch to the investment strategythat estimates correctly. 25
  26. 26. Order Price and Buy or Sell price Pet, j + Pd P t Sell (one unit) Pot, j > Pet, j Order Price Random Expected Price t t P Uniform o , j (about±10%) P e, j Buy (one unit) P t < P t o, j e, j Pet, j − Pd To Stabilize simulation for continuous double mechanism, Order Prices must be covered widely in Order Book. 26Next, agents determine order price and, buy or sell.To Stabilize simulation runs for the continuous double mechanism,Order Prices must be covered widely in Order Book.We modeled an Order Price, Po, by Random variables of Uniformly distributed inthe interval from Expected Price, Pe, minus constant, Pd, to Pe plus Pd.And then,When Po lager than Pe, the agent orders to sell one unit.When Po smaller than Pe, the agent orders to buy one unit. 26
  27. 27. Hazard Rate case 2 lerning non Price Limit Price Limit i 1 55% 55% 2 50% 53% 3 45% 49% 4 40% 47% hazard rate 5 36% 44% 6 29% 42% 7 26% 40% 8 22% 36% 9 19% 31% Hazard Rates increased → Result by preventing bubble 27Hazard RateHazard Rates increased in implemented price variation limitThis Result shows preventing bubble 27
  28. 28. Strategies Weight case 2, learning 10% 100% 9% 98% 8% 96% fundamental weight technical weight 7% 94% 6% 92% 5% 90% 4% 88% 3% 86% 2% fundamental weight (left axis) 84% 1% technical weight (right axis) 82% 0% 80% 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 time During overshooting, switching fundamental to technical Consistent with empirical studies 28 [Frankel and Froot, 1990], [Yamamoto and Hirata, 2012]I also show Strategies weight, in the case 2 with learningwhich case include overshoot.During overshooting, agents are switching fundamental strategy to technicalstrategy.This is consistent with empirical studies, like these studies. 28
  29. 29. Strategies Weight case 2, learning, Price Limit 10% 100% 9% 8% 95% fundamental weight technical weight 7% 6% 90% 5% 4% 85% 3% fundamental weight (left axis) 2% 80% 1% technical weight (right axis) 0% 75% 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 time Agents Not switching to the technical strategy → prevented overshooting. 29I also show Strategies weight, in the case 2 with learning, implemented pricevariation limitAgents are not switching fundamental strategy to technical strategy.This causes preventing overshooting, bubble and crash. 29

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